Number Theory, Homological Algebra
A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings
Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 33-39.

We prove Gersten’s conjecture for étale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As an application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.

Nous montrons la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux réguliers henséliens sans supposer de caractère équicaractéristique. En application, nous obtenons le principe local-global pour la cohomologie de Galois sur des anneaux locaux henséliens à deux dimensions de caractéristique mixte.

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DOI: 10.5802/crmath.9
Sakagaito, Makoto 1

1 Indian Institute of Science Education and Research, Mohali
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Sakagaito, Makoto. A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 33-39. doi : 10.5802/crmath.9. http://archive.numdam.org/articles/10.5802/crmath.9/

[1] Artin, Michael Grothendieck Topologies, Harvard University, 1962 | Zbl

[2] Bloch, Spencer; Ogus, Arthur Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér., Volume 7 (1974), pp. 181-201 | DOI | MR | Zbl

[3] Colliot-Thélène, Jean-Louis Quelques problèmes locaux-globaux (2011) (personal notes)

[4] Colliot-Thélène, Jean-Louis; Hoobler, Raymond T.; Kahn, Bruno The Bloch–Ogus–Gabber theorem, Algebraic K-theory (Fields Institute Communications), Volume 16, American Mathematical Society, 1997, pp. 31-94 | MR | Zbl

[5] Fujiwara, Kazuhiro A proof of the absolute purity conjecture (after Gabber), Algebraic geometry 2000, Azumino (Hotaka) (Advanced Studies in Pure Mathematics), Volume 36, Mathematical Society of Japan, 2000, pp. 153-183 | MR | Zbl

[6] Geisser, Thomas Motivic cohomology over Dedekind rings, Math. Z., Volume 248 (2004) no. 4, pp. 773-794 | DOI | Zbl

[7] Harbater, David; Hartmann, Julia; Krashen, Daniel Local-global principles for Galois cohomology, Comment. Math. Helv., Volume 89 (2014) no. 1, pp. 215-253 | DOI | MR | Zbl

[8] Hu, Yong A Cohomological Hasse Principle Over Two-dimensional Local Rings, Int. Math. Res. Not., Volume 2017 (2017) no. 14, pp. 4369-4397 | MR | Zbl

[9] Milne, James S. Étale Cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980 | MR | Zbl

[10] Panin, Ivan A. The equicharacteristic case of the Gersten conjecture, Tr. Mat. Inst. Im. V. A. Steklova, Volume 241 (2003) no. 2, pp. 169-178 | MR | Zbl

[11] Saito, Shuji Arithmetic on two-dimensional local rings, Invent. Math., Volume 85 (1986), pp. 379-414 | DOI | MR | Zbl

[12] Sakagaito, Makoto On problems about a generalization of the Brauer group (2016) (https://arxiv.org/abs/1511.09232v2)

[13] Sakagaito, Makoto On a generalized Brauer group in mixed characteristic cases (2019) (https://arxiv.org/abs/1710.11449v2)

[14] Voevodsky, Vladimir On motivic cohomology with Z/l-coefficients, Ann. Math., Volume 174 (2011) no. 1, pp. 401-438 | MR | Zbl

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